Doctoral Degrees (Applied Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7094

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Differential equations for relativistic radiating stars.
(2013) Abebe, Gezahegn Zewdie.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.
We consider radiating spherical stars in general relativity when they are conformally flat, geodesic with shear, and accelerating, expanding and shearing. We study the junction conditions relating the pressure to the heat flux at the boundary of the star in each case. The boundary conditions are nonlinear partial differential equations in the metric functions. We transform the governing equations to ordinary differential equations using the geometric method of Lie. The Lie symmetry generators that leave the equations invariant are identified, and we generate the optimal system in each case. Each element of the optimal system is used to reduce the partial differential equations to ordinary differential equations which are further analyzed. As a result, particular solutions to the junction conditions are presented for all types of radiating stars. New exact solutions, which are group invariant under the action of Lie point infinitesimal symmetries, are found. Our solutions contain families of traveling wave solutions, self-similar variables, and other forms with different combinations of the spacetime variables. The gravitational potentials are given in terms of elementary functions, and the line elements can be given explicitly in all cases. We show that the Friedmann dust model is regained as a special case in particular solutions. We can connect our results to earlier investigations and we show explicitly that our models are generalizations. Some of our solutions satisfy a linear equation of state. We also regain previously obtained solutions for the Euclidean star as a special case in our accelerating model. Our results highlight the importance of Lie symmetries of differential equations for problems arising in relativistic astrophysics.
Exact models of compact stars with equations of state.
(2013) Takisa, Pedro Mafa.; Maharaj, Sunil Dutt.
We study exact solutions to the Einstein-Maxwell system of equations and relate them to compact objects. It is well known that there are substantial analytic difficulties in the modelling of self-gravitating, static fluid spheres when the pressure explicitly depends on the matter density. Much simplification in solving the Einstein-Maxwell equations is achieved with the introduction of electric charge and anisotropic matter. In this thesis, in order to obtain analytical solutions, we consider the general situation of anisotropy in the presence of electric charge satisfying a barotropic equation of state. Firstly, a linear equation of state, secondly a quadratic equation of state, and thirdly a polytropic equation of state are analysed. For each of these equations of state the Einstein-Maxwell equations are integrated and exact solutions are found in terms of elementary functions. By choosing specific rational forms for one of the gravitational potential and particular forms for the electric charge, new classes of solutions in static spherically symmetric interior spacetimes are generated in the presence of electric charge. It is interesting to note that, from our new class of solutions with an equation of state, we can regain earlier models. A detailed physical analysis performed indicates that the classes of solutions are physically reasonable. We regain the current accurate observed masses for the binary pulsars PSR J1614-2230 , PSR J1903+327, Vela X-1, SMC X-1 and Cen X-3.
Group theoretic approach to heat conducting gravitating systems.
(2013) Nyonyi, Yusuf.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.
We study shear-free heat conducting spherically symmetric gravitating fluids defined in four and higher dimensional spacetimes. We analyse models that are both uncharged and charged via the pressure isotropy condition emanating from the Einstein field equations and the Einstein-Maxwell system respectively. Firstly, we consider the uncharged model defined in higher dimensions, and we use the algorithm due to Deng to generate new exact solutions. Three new metrics are identified which contain the results of four dimensions as special cases. We show graphically that the matter variables are well behaved and the speed of sound is causal. Secondly, we use Lie's group theoretic approach to study the condition of pressure isotropy of a charged relativistic model in four dimensions. The Lie symmetry generators that leave the equation invariant are found. We provide exact solutions to the gravitational potentials using the symmetries admitted by the equation. The new exact solutions contain earlier results without charge. We show that new charged solutions related to the Lie symmetries, that are generalizations of conformally at metrics, may be generated using the algorithm of Deng. Finally, we extend our study to find models of charged gravitating fluids defined in higher dimensional manifolds. The Lie symmetry generators related to the generalized pressure isotropy condition are found, and exact solutions to the gravitational potentials are generated. The new exact solutions contain earlier results obtained in four dimensions. Using particular Lie generators, we are able to provide forms for the gravitational potentials or reduce the order of the master equation to a first order nonlinear differential equation. Exact expressions for the temperature pro les, from the transport equation for both the causal and noncausal cases, in higher dimensions are obtained, generalizing previous results. In summary, the Deng algorithm and Lie analysis prove to be useful approaches in generating new models for gravitating fluids.
Relativistic radiating stars with generalised atmospheres.
(2010) Govender, Gabriel.; Maharaj, Sunil Dutt.
In this dissertation we construct radiating models for dense compact stars in relativistic astrophysics. We first utilise the standard Santos (1985) junction condition to model Euclidean stars. By making use of the heuristic Euclidean condition and a linear transformation in the gravitational potentials, we generate a particular exact solution in closed form to the nonlinear stellar boundary condition. Earlier models of spherical nonadiabatic gravitational collapse are then extended by considering the effect of radial perturbations in the matter and metric variables, on the evolution of the stellar fluid and the dynamics of the collapse process. The governing equation describing the temporal behaviour of the model is solved on the stellar surface. The model becomes static in the later stages of collapse. The Santos junction condition is then generalised to describe a radiating star which has a two-fluid atmosphere, consisting of a radiation field and a string fluid. We show that in the appropriate limit when the string energy density goes to zero, the standard result is regained. An exact solution to the generalised boundary condition is found. The generalised boundary condition is extended to hold in the case when the shear is nonvanishing. We demonstrate that our results can be used to model the flow of a string fluid in terms of a diffusion transport process.
Thermal evolution of radiation spheres undergoing dissipative gravitational collapse.
(2014) Reddy, Kevin Poobalan.; Govender, Megandren.; Maharaj, Sunil Dutt.
In this study we investigate the physics of a relativistic radiating star undergoing dissipative collapse in the form of a radial heat flux. Our treatment clearly demonstrates how the presence of shear affects the collapse process; we are in a position to contrast the physical features of the collapsing sphere in the presence of shear with the shear-free case. We first consider a particular exact solution found by Thirukkanesh et al [1] which is expanding, accelerating and shearing. By employing a causal heat transport equation of the Maxwell-Cattaneo form we show that the shear leads to an enhancement of the core stellar temperature thus emphasizing that relaxational effects cannot be ignored when the star leaves hydrostatic equilibrium. We also employ a perturbative scheme to study the evolution of a spherically symmetric stellar body undergoing gravitational collapse. The Bowers and Liang [2] static model is perturbed, and its subsequent dynamical collapse is studied in the linear perturbative regime. We find that anisotropic effects brought about by the differences in the radial and tangential pressures enhance the perturbations to the temperature, and that causal and non–causal cases yield identical profiles.

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Browsing Doctoral Degrees (Applied Mathematics) by Subject "Astrophysics."